Strong convexity for harmonic functions on compact symmetric spaces
- Resource Type
- Authors
- Dan Mangoubi; Zachary McGuirk; Gabor Lippner; Rachel Yovel
- Source
- Subject
- Physics
Mathematics - Differential Geometry
Pure mathematics
Applied Mathematics
General Mathematics
Operator (physics)
Function (mathematics)
Riemannian manifold
Type (model theory)
Convexity
Mathematics - Spectral Theory
Primary 43A85, Secondary 31C05, 22E30
Mathematics - Analysis of PDEs
Differential Geometry (math.DG)
Harmonic function
Homogeneous
Symmetric space
FOS: Mathematics
Spectral Theory (math.SP)
Analysis of PDEs (math.AP)
- Language
- English
Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on $\mathbb{R}^n$ discovered by the first two authors and is related to strong convexity of the $L^2$-growth function of harmonic functions.
Comment: 9 pages